Beyond Ability: The Four-Fold Spectrum of Power and the Logic of Full Inability

Abstract

Coalition Logic studies what coalitions can enforce. Recent work treats inability as simple non-ability: $\neg\Eff{C}\varphi$. This conflates two distinct configurations -- a coalition unable to force $\varphi$ may still force $\neg\varphi$, retaining adversarial control rather than genuine inability. We introduce \textbf{Full Inability} ($\FI$): the symmetric condition in which a coalition can enforce neither a proposition nor its negation. Combining coalitional effectivity with propositional negation yields a four-fold spectrum: \textbf{Full Control} ($\FC$), \textbf{Positive Determination} ($\PD$), \textbf{Adverse Determination} ($\AD$), and \textbf{Full Inability} ($\FI$). These categories partition a coalition's strategic status exhaustively and exclusively. We establish their algebraic and order-theoretic structure. Under $\alpha$-duality, propositional negation and coalition complementation generate a Klein four-group symmetry. In playable models, the four power regions are order-convex in the powerset lattice, yielding interval-stable verification of inability. We axiomatize $\CLFI$, a definitional extension treating Full Inability as a primitive modality. Via elimination translation, we prove soundness, completeness, and conservativity over Coalition Logic. The extension preserves expressive power and complexity ($\PSPACE$-complete), but provides direct proof-theoretic access to symmetric inability, strategic dependence, propositional dummyhood, and containment verification.